Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2009, Article ID 621625, 16 pages
doi:10.1155/2009/621625
Research Article
Warped Product Semi-Invariant Submanifolds in
Almost Paracontact Riemannian Manifolds
Mehmet Atçeken
Department of Mathematics, Faculty of Arts and Sciences, Gaziosmanpaşa University, 60250 Tokat, Turkey
Correspondence should be addressed to Mehmet Atçeken, matceken@gop.edu.tr
Received 2 December 2008; Revised 5 May 2009; Accepted 14 July 2009
Recommended by Fernando Lobo Pereira
We show that there exist no proper warped product semi-invariant submanifolds in almost
paracontact Riemannian manifolds such that totally geodesic submanifold and totally umbilical
submanifold of the warped product are invariant and anti-invariant, respectively. Therefore, we
consider warped product semi-invariant submanifolds in the form N N⊥ ×f NT by reversing
two factor manifolds NT and N⊥ . We prove several fundamental properties of warped product
semi-invariant submanifolds in an almost paracontact Riemannian manifold and establish a
general inequality for an arbitrary warped product semi-invariant submanifold. After then,
we investigate warped product semi-invariant submanifolds in a general almost paracontact
Riemannian manifold which satisfy the equality case of the inequality.
Copyright q 2009 Mehmet Atçeken. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
1. Introduction
It is well known that the notion of warped products plays some important role in differential
geometry as well as physics. The geometry of warped product was introduced by Bishop
and O’Neill 1. Many geometers studied different objects/structures on manifolds endowed
with an warped product metric see 2–6.
Recently, Chen has introduced the notion of CR-warped product in Kaehlerian
manifolds and showed that there exist no proper warped product CR-submanifolds in the
form N N⊥ ×f NT in Kaehlerian manifolds. Therefore, he considered warped product CRsubmanifolds in the form N NT ×f N⊥ which is called CR-warped product, where NT is
an invariant submanifold, and N⊥ is an anti-invariant submanifold of Kaehlerian manifold
M see 2, 7, 8. Analogue results have been obtained for Sasakian ambient as the odd
dimensional version of Kaehlerian manifold by Hasegawa and Mihai in 3 and Munteanu
in 9.
Almost paracontact manifolds and almost paracontact Riemannian manifolds were
defined and studied by Şato 10. After then, many authors studied invariant and
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anti-invariant submanifolds of the almost paracontact Riemannian manifold M with the
structure F, g, ξ, η, when ξ is tangent to the submanifold, and ξ is not tangent to the
submanifold 11.
We note that CR-warped products in Kaehlerian manifold are corresponding warped
product semi-invariant submanifolds in almost paracontact Riemannian manifolds. In this
paper, we showed that there exist no warped product semi-invariant submanifolds in the
form N NT ×f N⊥ in contrast to Kaehlerian manifolds see Theorem 3.1. So, from now
on we consider warped product semi-invariant submanifolds in the form N N⊥ ×f NT ,
where N⊥ is an anti-invariant submanifold, and NT is an invariant submanifold of an almost
paracontact Riemannian manifold M by reversing the two factor manifolds NT and N⊥ , and
it simply will be called warped product semi-invariant submanifold in the rest of this paper
see Example 3.3 and Theorem 3.4.
2. Preliminaries
Although there are many papers concerning the geometry of semi-invariant submanifolds
of almost paracontact Riemannian manifolds see 11–13, there is no paper concerning the
geometry of warped product semi-invariant submanifolds of almost paracontact Riemannian
manifolds in literature so far. So the purpose of the present paper is to study warped
product semi-invariant submanifolds in almost paracontact Riemannian manifolds. We first
review basic formulas and definitions for almost paracontact Riemannian manifolds and their
submanifolds, which we shall use for later.
Let M be an m 1-dimensional differentiable manifold. If there exist on M a 1, 1
type tensor field F, a vector field ξ, and 1-form η satisfying
F 2 I − η ⊗ ξ,
ηξ 1,
2.1
then M is said to be an almost paracontact manifold, where ⊗, the symbol, denotes the tensor
product. In the almost paracontact manifold, the following relations hold good:
Fξ 0,
η ◦ F 0,
rankF m.
2.2
An almost paracontact manifold M is said to be an almost paracontact metric manifold
if Riemannian metric g on M satisfies
gFX, FY gX, Y − ηXηY ,
ηX gX, ξ
2.3
for all X, Y ∈ ΓT M 14, where ΓT M denotes the differentiable vector field set on M.
From 2.2 and 2.3, we can easily derive the relation
gFX, Y gX, FY .
2.4
An almost paracontact metric manifold is said to be an almost paracontact Riemannian
manifold with F, g, ξ, η-connection if ∇F 0 and ∇η 0, where ∇ denotes the connection
on M. Since F 2 I − η ⊗ ξ, the vector field ξ is also parallel with respect to ∇ 11, 13.
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In the rest of this paper, let us suppose that M is an almost paracontact Riemannian
manifold with structure F, g, ξ, η-connection.
Let M be an almost paracontact Riemannian manifold, and let N be a Riemannian
manifold isometrically immersed in M. For each x ∈ N, we denote by Dx the maximal
invariant subspace of the tangent space Tx N of N. If the dimension of Dx is the same for
all x in N, then Dx gives an invariant distribution D on N.
A submanifold N in an almost paracontact Riemannian manifold is called semiinvariant submanifold if there exists on N a differentiable invariant distribution D whose
orthogonal complementary D⊥ is an anti-invariant distribution, that is, FD⊥ ⊂ T N ⊥ , where
T N ⊥ denotes the orthogonal vector bundle of T N in T M. A semi-invariant submanifold is
called anti-invariant resp., invariant submanifold if dimDx 0 resp., dimDx⊥ 0.
It is called proper semi-invariant submanifold if it is neither invariant nor anti-invariant
submanifold.
A semi-invariant submanifold N of an almost paracontact Riemannian manifold M is
called a Riemannian product if the invariant distribution D and anti-invariant distribution
D⊥ are totally geodesic distributions in N. The geometry notion of the semi-invariant
submanifolds has been studied by many geometers in the various type manifolds. Authors
researched the fundamental properties of such submanifolds see references.
Let N1 and N2 be two Riemannian manifolds with Riemannian metrics g1 and g2 ,
respectively, and f a differentiable function on N1 . Consider the product manifold N1 × N2
with its projection π1 : N1 ×N2 → N1 and π2 : N1 ×N2 → N2 . The warped product manifold
N N1 ×f N2 is the manifold N1 × N2 equipped with the Riemannian metric tensor such that
gX, Y g1 π1∗ X, π1∗ Y f 2 π1 xg2 π2∗ X, π2∗ Y 2.5
for any X, Y ∈ ΓT N, where ∗ is the symbol for the tangent map. Thus we have g g1 f 2 g2 ,
where f is called the warping function of the warped product. The warped product manifold
N N1 ×f N2 is characterized by the fact that N1 and N2 are totally geodesic and totally
umbilical submanifolds of N, respectively. Hence Riemannian products are special classes of
the warped products 4.
In this paper, we define and study a new class of warped product semi-invariant
submanifolds in an almost paracontact Riemannian manifolds, namely, we investigate the
class of warped product semi-invariant submanifolds, and we establish the fundamental
theory of such submanifolds.
Now, let N be an isometrically immersed submanifold in an almost paracontact
Riemannian manifold M. We denote by ∇ and ∇ the Levi-Civita connections on N and M,
respectively. Then the Gauss and Weingarten formulas are, respectively, defined by
∇X Y ∇X Y hX, Y ,
∇X V −AV X ∇⊥X V
2.6
for any X, Y ∈ ΓT N, V ∈ ΓT N ⊥ , where ∇⊥ is the connection in the normal bundle T N ⊥ , h
is the second fundamental form of N, and AV is the shape operator. The second fundamental
form h and the shape operator A are related by
gAV X, Y ghX, Y , V .
2.7
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Now, let N be a differentiable manifold, and we suppose that N is an isometrically
immersed submanifold in almost paracontact Riemannian manifold M. We denote by g the
metric tensor of M as well as that induced on N. For any vector field X tangent to N, we put
FX tX nX,
2.8
where tX and nX denote the tangential and normal components of FX, respectively. For any
vector field V normal to N, we also put
FV BV CV,
2.9
where BV and CV denote the tangential and normal components of FV , respectively. The
submanifold N is said to be invariant if n is identically zero, that is, FT N T N. On the
other hand, N is said to be anti-invariant submanifold if t is identically zero, that is, FT N ⊂
T N ⊥ .
We note that for any invariant submanifold N of an almost paracontact Riemannian
manifold M, if ξ is normal to N, then the induced structure from the almost paracontact
structure on N is an almost product Riemannian structure whenever t is nontrivial. If ξ is
tangent to N, then the induced structure on N is an almost paracontact Riemannian structure.
Furthermore, we say that N is a semi-invariant submanifold if there exist two
orthogonal distributions D1 and D2 such that
1 T N splits into the orthogonal direct sum T N D1 ⊕ D2 ;
2 the distribution D1 is invariant, that is, FD1 ⊆ D1 ;
3 the distribution D2 is anti-invariant, that is, FD2 ⊆ T N ⊥ .
Given any submanifold N of an almost paracontact Riemannian manifold M, from
2.4 and 2.8 we have
gtX, Y gX, tY ,
gnX, V gX, BV 2.10
for any X, Y ∈ ΓT N, V ∈ ΓT N ⊥ .
From now on we suppose that the vector field ξ is tangent to N.
We recall the following general lemma from 1 for later use.
Lemma 2.1. Let N N1 ×f N2 be a warped product manifold with warping function f, then one has
1 ∇X Y ∈ ΓT N1 for each X, Y ∈ ΓT N1 ,
2 ∇X Z ∇Z X Xln fZ, for each X ∈ ΓT N1 , Z ∈ ΓT N2 ,
2
3 ∇Z W ∇N
Z W − gZ, Wgradf/f, for each Z, W ∈ ΓT N2 ,
where ∇ and ∇N2 denote the Levi-Civita connections on N and N2 , respectively.
Let N be a semi-invariant submaniold of an almost paracontact Riemannian manifold
M. We denote by the invariant distribution D and anti-invariant distribution D⊥ . We also
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denote the orthogonal complementary of FD⊥ in T N ⊥ by ν, then we have direct sum
T N ⊥ F D⊥ ⊕ ν.
2.11
We can easily see that ν is an invariant subbundle with respect to F.
3. Warped Product Semi-Invariant Submanifolds in
an Almost Paracontact Riemannian Manifold
Useful characterizations of warped product semi-invariant submanifolds in almost paracontact Riemannian manifolds are given the following theorems.
Theorem 3.1. If N NT ×f N⊥ is a warped product semi-invariant submanifold of an almost
paracontact Riemannian manifold M such that NT is an invariant submanifold and N⊥ is an antiinvariant submanifold of M, then N is a usual Riemannian product.
Proof. Let ξ be normal to N. Taking into account that h is symmetric and using 2.3, 2.6,
2.7, and considering Lemma 2.12, for X ∈ ΓT NT and Z, W ∈ ΓT N⊥ , we have
g∇X Z, W g∇Z X, W g ∇Z X, W g F∇Z X, FW η ∇Z X ηW,
X ln f gZ, W g ∇Z FX, FW ghZ, FX, FW g ∇FX Z, FW
g ∇FX FZ, W −gAFZ FX, W −ghFX, W, FZ
3.1
−g ∇W FX, FZ −g ∇W X, Z −X ln f gW, Z,
which implies that X lnf 0.
If ξ is tangent to N, then ξ can be written as follows:
ξ ξ1 ξ2 ,
3.2
where ξ1 ∈ ΓT NT and ξ2 ∈ ΓT N⊥ . Since ∇X ξ 0, from the Gauss formulae, we have
hX, ξ 0,
∇X ξ 0
3.3
for any X ∈ ΓT N. Considering Lemma 2.12, we get
∇X ξ2 X ln f ξ2 0,
∇ξ2 ξ1 ∇ξ1 ξ2 ξ1 ln f ξ2 0
∇Z ξ1 ξ1 ln f Z 0,
3.4
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for any X ∈ ΓT NT and Z ∈ ΓT N⊥ . If ξ2 is identically zero, then from Lemma 2.1 we have
∇Z ξ1 ∇ξ1 Z ξ1 ln f Z 0,
∇X ξ1 ∈ ΓT NT .
3.5
It follows that the warping function f is a constant and N is usual Riemannian product.
Hence the proof is complete.
If the warping function is constant, then the metric on the “second” factor could be
modified by an homothety, and hence, the warped product becomes a direct product.
Now, we give two examples for almost paracontact Riemannian manifold and their
submanifolds in the form N N⊥ ×f NT to illustrate our results. Firstly, we construct an
almost paracontact metric structure on R2n1 see Example 3.2 and after give an example
which is concerning its submanifold see Example 3.3.
Example 3.2. Let
R2n1 x1 , x2 , . . . , xn , y1 , y2 , . . . , yn , t | xi , yi , t ∈ R, i 1, 2, . . . , n .
3.6
The almost paracontact Riemannian structure F, g, ξ, η is defined on R2n1 in the following
way:
F
∂
∂xi
∂
,
∂yi
F
∂
∂yi
∂
,
∂xi
F
∂
∂t
0,
ξ
∂
,
∂t
η dt.
3.7
If Z λi ∂/∂xi μi ∂/∂yi ν∂/∂t ∈ T R2n1 , then we have
gZ, Z n
i1
λ2i n
μ2i ν2 .
3.8
i1
From this definition, it follows that
gZ, ξ ηZ ν,
gFZ, FZ gZ, Z − η2 Z,
Fξ 0,
ηξ 1
3.9
for an arbitrary vector field Z. Thus R2n1 , F, g, ξ, η becomes an almost paracontact
Riemannian manifold, where g and {∂/∂xi , ∂/∂yi , ∂/∂t} denote usual inner product and
standard basis of T R2n1 , respectively.
Example 3.3. Let N be a submanifold in R5 with coordinates x1 , x2 , y1 , y2 , t given by
x1 v cos θ,
x2 v sin θ,
y1 v cos β,
y2 v sin β,
t
√
2u.
3.10
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It is easy to check that the tangent bundle of N is spanned by the vectors
Z1 cos θ
∂
∂
∂
∂
sin θ
cos β
sin β
,
∂x1
∂x2
∂y1
∂y2
Z2 −v sin θ
∂
∂
v cos θ
,
∂x1
∂x2
3.11
∂
∂
Z3 −v sin β
v cos β
,
∂y1
∂y2
Z4 √ ∂
2 .
∂t
We define the almost paracontact Riemannian structure of R5 by
F
∂
∂xi
−
∂
,
∂xi
F
∂
∂yi
∂
,
∂yi
F
∂
∂t
0,
1
η √ dt.
2
3.12
Then with respect to the almost paracontact Riemannian structure F of R5 , the space FT N
becomes
FZ1 − cos θ
∂
∂
∂
∂
− sin θ
cos β
sin β
,
∂x1
∂x2
∂y1
∂y2
FZ2 v sin θ
∂
∂
− v cos θ
,
∂x1
∂x2
FZ3 −v sin β
3.13
∂
∂
v cos β
,
∂y1
∂y2
FZ4 0.
Since FZ1 and FZ4 are orthogonal to N and FZ2 , FZ3 are tangent to N, D and D⊥ can be
taken subspace sp{Z1 , Z4 } and subspace sp{Z2 , Z3 }, respectively, where ξ can be taken as Z4
for FZ4 0 and ηZ4 1. Furthermore, the metric tensor of N is given by
g 2 du2 dv2 v2 dθ2 dβ2 2gN⊥ v2 gNT .
3.14
Thus N is a warped product semi-invariant submanifold with dimensional 5 of almost
paracontact manifold R5 with warping function f v2 .
Now, let N N⊥ ×f NT be a warped product semi-invariant submanifold of an almost
paracontact Riemannian manifold M, where N⊥ is an anti-invariant submanifold, and NT is
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an invariant submanifold of M. If we denote the Levi-Civita connections on M and N by ∇
and ∇, respectively, by using 2.6 and 2.8, we have
∇X FY F∇X Y,
∇X tY hX, tY − AnY X ∇⊥X nY t∇X Y n∇X Y BhX, Y ChX, Y 3.15
for any X, Y ∈ ΓT N. Taking into account the tangential and normal components of 3.15,
respectively, we have
∇X tY AnY X BhX, Y ,
3.16
∇X nY ChX, Y − hX, tY ,
3.17
where the derivatives of t and n are, respectively, defined by
∇X tY ∇X tY − t∇X Y ,
∇X nY ∇⊥X nY − n∇X Y .
3.18
Next, we are going to investigate the geometric properties of the leaves of the warped product
semi-invariant submanifolds in an almost paracontact Riemannian manifold.
Theorem 3.4. Let N be a warped product semi-invariant submanifold of an almost paracontact
Riemannian manifold M. Then the invariant distribution D and the anti invariant distribution D⊥
are always integrable.
Proof. From 3.16 and considering Lemma 2.11, we have
∇X FU F∇X U,
X ln f tU hX, tU X ln f tU BhX, U ChX, U
3.19
for any X ∈ ΓD⊥ and U ∈ ΓD. From the tangential and normal components of 3.19,
respectively, we arrive at
BhX, U 0,
3.20
hX, tU ChX, U.
3.21
AnX U −X ln f tU.
3.22
By using 3.16 and 3.20 we get
Furthermore, by using the Gauss-Weingarten formulas and taking into account that D⊥ is
totally geodesic in N and it is anti-invariant in M, by direct calculations, it is easily to see
that
AnY X −BhX, Y ,
3.23
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which is also equivalent to
AnY X AnX Y
3.24
for any X, Y ∈ ΓD⊥ . Moreover, using 2.4 and 2.7 and making use of A being self-adjoint,
we obtain
gAnX Y, Z ghY, Z, nX g ∇Z Y, FX g ∇Z FY, X
3.25
−gAnY Z, X −gAnY X, Z,
which gives us
AnX Y −AnY X
3.26
for any X, Y ∈ ΓD⊥ and Z ∈ ΓT N. Thus from 3.24 and 3.26, we arrive at
AnX Y 0,
BhX, Y 0
3.27
for any X, Y ∈ ΓD⊥ . Furthermore, by using 2.6, 2.8, and 2.9 and considering
Lemma 2.13, we have
hU, tV ∇U tV F∇U V FhV, U
gradf
BhV, U ChV, U
F ∇U V − gV, U
f
gradf
t ∇U V − gV, Un
BhV, U ChV, U
f
3.28
for any V, U ∈ ΓD, where ∇ denote the Levi-Civita connection on D. Taking into account
the normal and tangential components of 3.28, respectively, we have
gradf
hU, tV −gU, V n
ChU, V ,
f
∇U tV − gtV, U
gradf
t ∇U V BhV, U.
f
3.29
3.30
From 3.29, we can easily see that
hU, tV hV, tU
3.31
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for any U, V ∈ ΓD. Finally, by using 3.17 and 3.31, we have
nV, U n∇V U − ∇U V ∇⊥V nU − ∇V nU − ∇⊥U nV ∇U nV
∇U nV − ∇V nU ChU, V − hU, tV − ChV, U hV, tU 0
3.32
for any V, U ∈ ΓD, that is, V, U ∈ ΓD.
In the same way, making use of 3.16 and 3.27 for any X, Y ∈ ΓD⊥ , we conclude
that
tX, Y t∇X Y − ∇Y X
∇X tY − ∇X tY − ∇Y tX ∇Y tX
3.33
∇Y tX − ∇X tY AnX Y − AnY X 0
that is, X, Y ∈ ΓD⊥ . So we obtain the desired result.
Since the distributions D and D⊥ are integrable, we denote the integral manifolds of D
and D⊥ by NT and N⊥ , respectively.
Now, the following theorem characterizes warped product or Riemannian product
semi-invariant submanifolds in almost paracontact manifolds.
Theorem 3.5. Let N be a submanifold of an almost paracontact Riemannian manifold M. Then N is
a semi-invariant submanifold if and only nt 0.
Proof. Let us assume that N is a semi-invariant submanifold of an almost paracontact
Riemannian manifold M and by Q and P ; we denote the projection operators on subspaces
ΓD⊥ and ΓD, respectively, then we have
P Q I,
P 2 P,
Q2 Q,
P Q QP 0.
3.34
Moreover, by using 2.1, 2.8, and 2.9, if ξ is tangent to N, then we get
X − ηXξ t2 X BnX,
tBV BCV 0,
ntX CnX 0,
nBV C2 V V
3.35
3.36
for any X ∈ ΓT N and V ∈ ΓT N ⊥ . On the other hand, if ξ is normal to N, then 3.35 and
3.36 become, respectively,
X t2 X BnX,
V − ηV ξ nBV C2 V,
ntX CnX 0,
tBV BCV 0.
3.37
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From 2.8, we have
FX FP X FQX,
tX nX tP X tQX nP X nQX
3.38
for any X ∈ ΓT N. From the tangential and normal components, we have
tX tP X tQX,
nX nP X nQX.
3.39
Since D is invariant and D⊥ is anti-invariant, we get
nP 0,
Qt 0.
3.40
We have
tP t
3.41
by virtue of Q I − P . Now by using the right-hand side to the second equation of 3.35 and
using 3.40 and 3.41, we conclude that
nt 0,
3.42
Cn 0.
3.43
which is also equivalent to
Conversely, for a submanifold N of an almost paracontact Riemannian manifold M,
we suppose that nt 0. For any vector fields tangent X to N and V normal to N, by using
2.4 and 3.43, we have
gX, FV gFX, V ,
gX, BV gnX, V ,
gX, FBV gFnX, V ,
3.44
gX, tBV gCnX, V 0
for all X ∈ ΓT N. So we have gtBV, X 0. Since X, tBV ∈ ΓT N, it implies tB 0 which
is also equivalent to BC 0 from 3.36. Since Fξ 0, we get tξ nξ 0. So, from 3.35 and
3.36, we conclude
t3 t,
C3 C.
3.45
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Now, if we put
P t2 ,
3.46
Q I − P,
then we can derive that P Q I, P 2 P, Q2 Q, and P Q QP 0 which show
that Q and P are orthogonal complementary projection operators and define complementary
distributions D⊥ and D, respectively, where D and D⊥ denote the distributions which are
belong to subspaces T NT and T N⊥ , respectively. From 3.42, 3.45, and 3.46 we can derive
tP t,
tQ 0,
QtP 0,
nP 0.
3.47
These equations show that the distribution D is an invariant and the distribution D⊥ is an
anti-invariant. The proof is complete.
Theorem 3.6. Let N be a semi-invariant submanifold of an almost paracontact Riemannian manifold
M. Then N is a warped product semi-invariant submanifold if and only if the shape operator of N
satisfies
AFX Z −X μ FZ,
X ∈ Γ D⊥ , Z ∈ ΓD
3.48
for some function μ on N satisfying Wμ 0, W ∈ ΓD.
Proof. We suppose that N is a warped product semi-invariant submanifold in an almost
paracontact Riemannian manifold M. Then from 3.22, we have
AFX Z −X ln f FZ
3.49
for any X ∈ ΓD⊥ and Z ∈ ΓD. Since f is the only function on N⊥ , we can easily see that
Wln f 0 for all W ∈ ΓD.
Conversely, let us assume that N is a semi-invariant submanifold in an almost
paracontact Riemannian manifold M satisfying
AFX Z X μ FZ,
X ∈ Γ D⊥ , Z ∈ ΓD
3.50
for some function μ on N satisying Wμ 0 for all W ∈ ΓD. Since the ambient space M is
an almost paracontact Riemannian manifold and making use of 2.4 and 3.27, we arrive at
g∇X Y, FZ g ∇X Y, FZ g ∇X FY, Z −gAFY X, Z 0
3.51
for any X, Y ∈ ΓD⊥ and Z ∈ D. Thus the anti-invariant distribution D⊥ is totally geodesic
in N. In the same way, making use of ∇ being Levi-Civita connection and 3.22, we have
g∇Z W, X g ∇Z W, X −g ∇Z X, W −g ∇Z FX, FW
gAFX Z, FW X μ gZ, W
3.52
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for any Z, W ∈ ΓD and X ∈ ΓD⊥ , where μ ln1/f. Since the invariant distribution
D of semi-invariant submanifold N is always integrable Theorem 3.4 and Wμ 0, for
each W ∈ ΓT NT , which implies that the integral manifold of D is an extrinsic sphere in N,
that is, it is a totally umbilical submanifold and its mean curvature vector field is non-zero
and parallel, thus we know that N is a Riemannian warped product N⊥ ×f NT , where N⊥ and
NT denote the integral manifolds of the distributions of D⊥ and D, respectively, and f is the
warping function. So we obtain the desired result.
In the rest of this section, we are going to obtain an inequality for the squared norm
of the second fundamental form by means of the warping function for warped product
semi-invariant submanifolds of an almost paracontact Riemannian manifold. Now, we recall
that semi-invariant N is said to be mixed geodesic resp., D-geodesic and D⊥ -geodesic
submanifold if the second fundamental form h of N satisfies hX, Z 0, X ∈ ΓD and
Z ∈ ΓD⊥ resp., hX, Y 0, X, Y ∈ ΓD and hZ, W 0, Z, W ∈ ΓD⊥ .
Now, we are going to give the following lemma for later use.
Lemma 3.7. Let N N⊥ ×f NT be a warped product semi-invariant submanifold of an almost
paracontact Riemannian manifold M. Then one has
1 ghD⊥ , D⊥ , FD⊥ 0,
2 ghZ, W, FX −Xln fgtZ, W, Z, W ∈ ΓD, X ∈ ΓD⊥ ,
3 ghX, Z, FY 0, for any X, Y ∈ ΓD⊥ and Z ∈ ΓD,
4 ghD, FD, FD⊥ 0 if and only if N N⊥ ×f NT is a usual Riemannian product, where
D and D⊥ denote the leaves of NT and N⊥ , respectively.
Proof. 1 For any X, Y, Z ∈ ΓD⊥ , by using 2.4 and 3.27 and considering that the ambient
space is an almost paracontact Riemannian manifold, we have
ghX, Y , FZ g ∇X Y, FZ g ∇X FY, Z −gAnY X, Z 0.
3.53
2 Making use of ∇ being Levi-Civita connection and Lemma 2.12.2, we get
ghZ, W, FX g ∇W FZ, X −g ∇W X, tZ −X ln g · gW, tZ
3.54
for any Z, W ∈ ΓD, X ∈ ΓD⊥ .
3 In the same way, we have
ghX, Z, FY g ∇X Z, FY g∇X tZ, Y X ln fgtZ, Y 0
3.55
for any X, Y ∈ ΓD⊥ and Z ∈ ΓD.
4 Considering Lemma 2.13 we derive
ghW, FZ, FX g ∇FZ W, FX g ∇FZ FW, X −gtZ, tWX ln f
for anyZ, W ∈ ΓD and X ∈ ΓD⊥ .
3.56
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Mathematical Problems in Engineering
Theorem 3.8. Let N N⊥ ×f NT be a warped product semi-invariant submanifold of an almost
paracontact Riemannian manifold M. Then one has the following.
1 The squared norm of the second fundamental form of N in M satisfies
h2 ≥
1
gradf 2 Trt2 ,
2
f
3.57
where Trt denote the trace of mapping t.
2 If the equality sign of 3.57 holds identically, then N⊥ is a totally geodesic, NT is a totally
umbilical submanifolds of M, and N is a mixed geodesic submanifold in M. Furthermore,
N is a minimal submanifold M if and only if Trt 0 or N N⊥ ×f NT is a usual
Riemannian product.
Proof. Let {e1 , e2 , . . . , ep , e1 , e2 , . . . , eq , N1 , N2 , . . . , Ns , ξ} be an orthonormal basis of an
almost paracontact Riemannian manifold M such that {e1 , e2 , . . . , ep } is tangent to
ΓT N⊥ , {e1 , e2 , . . . , eq } is tangent to ΓT NT , and {N1 , N2 , . . . , Ns } is tangent to Γν. Taking
into account Lemma 3.7 and the basic linear algebra rules, by direct calculations, we have
hX, Y g hX, Y , Nj Nj ,
1 ≤ j ≤ s,
hZ, W −ei ln ftZ, WFei g hZ, W, Nj Nj ,
hX, Z g hX, Z, Nj Nj
1 ≤ i ≤ p,
3.58
for all X, Y ∈ ΓT N⊥ and Z, W ∈ ΓT NT . Since
h2 p p q s
s
2
2
g h ei , ej , N
2
g h ei , e k , N
i,j1 1
i1 k1 1
q s
2 2
2 g h e k , e r , N
,
ei ln f g tek , er q p
r,k1 i1
3.59
r,k1 1
here by direct calculations, we get
2
2
1
ei ln f
2 gradf ,
f
Trt q
g tek , ek .
3.60
k1
So we conclude that
h2 ≥
which proves our assertion.
1
gradf 2 Trt2 ,
2
f
3.61
Mathematical Problems in Engineering
15
Now we assume that the equality case of 3.57 holds identically, then from 3.58,
respectively, we obtain
h D⊥ , D⊥ 0, hD, D ∈ Γ F D⊥ ,
h D⊥ , D 0.
3.62
3.63
Since N⊥ is totally geodesic submanifold in N, the first condition in 3.62 implies that
N⊥ is totally geodesic submanifold in M. Moreover, Lemma 2.13 shows that NT is totally
umbilical submanifold in N. Therefore, the second condition in 3.62 implies that NT is also
totally umbilical submanifold in M. On the other hand, 3.20 and 3.63 imply that N is
mixed geodesic submanifold in M.
Conclusion 3.9. The geometry of the warped products in Riemannian manifolds is totally
different from the geometry of the warped products in complex manifolds. Namely, in the
complex manifolds, there exists no proper warped product CR-submanifold in the form N N⊥ ×f NT see 2, 8 while there exists no proper warped product semi-invariant submanifold
in the form N NT ×f N⊥ in Riemannian manifolds see Theorem 3.1. The first condition in
3.62 implies that warped product CR-submanifold is minimal in complex manifolds while
it does not imply that warped product semi-invariant submanifold is minimal in Riemannian
product manifolds.
Acknowledgment
The author would like to thank the referees for valuable suggestions and comments, which
have improved the present paper.
References
1 R. L. Bishop and B. O’Neill, “Manifolds of negative curvature,” Transactions of the American
Mathematical Society, vol. 145, pp. 1–49, 1969.
2 B.-Y. Chen, “Geometry of warped product CR-submanifolds in Kaehler manifolds,” Monatshefte für
Mathematik, vol. 133, no. 3, pp. 177–195, 2001.
3 I. Hasegawa and I. Mihai, “Contact CR-warped product submanifolds in Sasakian manifolds,”
Geometriae Dedicata, vol. 102, pp. 143–150, 2003.
4 V. A. Khan, K. A. Khan, and Sirajuddin, “Contact CR-warped product submanifolds of Kenmotsu
manifolds,” Thai Journal of Mathematics, vol. 6, no. 1, pp. 138–145, 2008.
5 K. Matsumoto and I. Mihai, “Warped product submanifolds in Sasakian space forms,” SUT Journal of
Mathematics, vol. 38, no. 2, pp. 135–144, 2002.
6 D. Won Yoon, K. Soo Cho, and S. Gook Han, “Some inequalities for warped products in locally
conformal almost cosymplectic manifolds,” Note di Matematica, vol. 23, no. 1, pp. 51–60, 2004.
7 B.-Y. Chen, “CR-warped products in complex projective spaces with compact holomorphic factor,”
Monatshefte für Mathematik, vol. 141, no. 3, pp. 177–186, 2004.
8 B.-Y. Chen, “Geometry of warped product CR-submanifolds in Kaehler manifolds. II,” Monatshefte für
Mathematik, vol. 134, no. 2, pp. 103–119, 2001.
9 M. I. Munteanu, “Warped product contact CR-submanifolds of Sasakian space forms,” Publicationes
Mathematicae Debrecen, vol. 66, no. 1-2, pp. 75–120, 2005.
10 I. Satō, “On a structure similar to the almost contact structure,” The Tensor Society. Tensor. New Series,
vol. 30, no. 3, pp. 219–224, 1976.
11 S. Ianus, K. Matsumoto, and I. Mihai, “Almost semi-invariant submanifolds of some almost
paracontact Riemannian manifolds,” Bulletin of Yamagata University, vol. 11, no. 2, pp. 121–128, 1985.
16
Mathematical Problems in Engineering
12 A. Bejancu, “Semi-invariant submanifolds of locally product Riemannian manifolds,” Analele,
Universitatii Din Timisoara, Seria Matematica-Informatica, vol. 22, no. 1-2, pp. 3–11, 1984.
13 S. Ianus and I. Mihai, “Semi-invariant submanifolds of an almost paracontact manifold,” The Tensor
Society. Tensor, vol. 39, pp. 195–200, 1982.
14 N. Jovanka, “Conditions for invariant submanifold of a manifold with the ϕ, ξ, η, G-structure,”
Kragujevac Journal of Mathematics, vol. 25, pp. 147–154, 2003.